H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = 0. It is initially at rest, with velocity u₁ = 0. The other end of the spring is attached to a fixed wall at position x = - where l is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m₂ travels towards particle 1 with constant velocity u₂ (note that u₂ < 0 in this setup). The two particles undergo a collision with coefficient of restitution 0≤e ≤ 1. (a) Using the one-dimensional equations of collision presented in the lectures show that imme- diately after collision the velocity of particle 1 is given by v₁ = (1+e); m242 m₁ + m₂ Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2 if m₁ = m₂ and the collision is elastic (e = 1)? (b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we take as initial conditions for its subsequent motion under the influence of the spring (we assume that there is no friction or resistance). Use energy arguments to determine how close particle 1 comes to the wall. In other words, find the minimum value of x(t) in terms of e, k, u2, m₁ and m₂. If you aim to send particle 1 as close as possible to the wall by acting only on the coefficient of restitution e, what value of e would you pick? (We implicitly assume that the length of the spring is sufficiently long to ensure that the previous answer makes sense, i.e., l is large enough so that particle 1 does not crash into the wall)

H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = 0. It is initially at rest, with velocity u₁ = 0. The other end of the spring is attached to a fixed wall at position x = - where l is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m₂ travels towards particle 1 with constant velocity u₂ (note that u₂ < 0 in this setup). The two particles undergo a collision with coefficient of restitution 0≤e ≤ 1. (a) Using the one-dimensional equations of collision presented in the lectures show that imme- diately after collision the velocity of particle 1 is given by v₁ = (1+e); m242 m₁ + m₂ Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2 if m₁ = m₂ and the collision is elastic (e = 1)? (b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we take as initial conditions for its subsequent motion under the influence of the spring (we assume that there is no friction or resistance). Use energy arguments to determine how close particle 1 comes to the wall. In other words, find the minimum value of x(t) in terms of e, k, u2, m₁ and m₂. If you aim to send particle 1 as close as possible to the wall by acting only on the coefficient of restitution e, what value of e would you pick? (We implicitly assume that the length of the spring is sufficiently long to ensure that the previous answer makes sense, i.e., l is large enough so that particle 1 does not crash into the wall)